Exact Controllability For Quasilinear Hyperbolic Systems

In this thesis, we establish the existence and uniqueness of semi-global Cl solution to the mixed initial-boundary value problem with nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues , and obtain the corresponding exact controllability. As applications, by means of boundary controls acting on one end or on two ends, we get the exact boundary controllability for the quasilinear wave equations with boundary conditions of various types, for second order quasilinear hyperbolic systems and for higher order quasilinear hyperbolic equations.The arrangement of this thesis is as follows:In Chapter 1, we give a brief introduction and the present situation on the exact controllability for hyperbolic systems (equations). The main results and methods with the sketch of proof are also presented.In Chapter 2, we prove the existence and uniqueness of semi-global Cl solution to the mixed initial-boundary value problem for first order quasilinear hyperbolic systems with zero eigenvalues.In Chapter 3, by means of the result obtained in Chapter 2, we realize the local exact controllability with boundary controls acting on one end or on two ends and internal controls acting on a part of equations corresponding to zero eigenvalues for quasilinear hyperbolic systems with zero eigenvalues.In Chapter 4, by the existence and uniqueness of semi-global Cl solution to the mixed initial-boundary value problem for first order quasilinear hyperbolic systems with zero eigenvalues obtained in Chapter 2, by means of a unified method we get the existence and uniqueness of semi-global C2 solution to the mixed initial-boundary value problem for the quasi-linear wave equations with various boundary conditions, and establish the exact boundary controllability for quasilinear wave equations with boundary conditions of various types.In Chapter 5, still using the result obtained in Chapter 2, we get the existence and uniqueness of semi-global C2 solution for second order quasilinear hyperbolic systems and obtain the exact boundary controllability for second order quasilinear hyperbolic systemswith general nonlinear boundary conditions.In Chapter 6, in a similar way, we obtain the exact boundary controllability for higher order quasilinear hyperbolic equations with general nonlinear boundary conditions.